$$​\frac{1}{3}$$ of a commodity is sold at the rate of 15% profit, $$\frac{1}{4}$$ is sold at the rate of 20% profit and the rest is sold at the rate of 24% profit. If a total profit of ₹62 is earned, then find the cost price of the commodity.

Correct option is A

Given:

$$\frac{1}{3}$$of the commodity sold at 15% profit

$$\frac{1}{4}$$ at 20% profit

Remaining at 24% profit

Total profit = ₹62

Solution:
Let total cost price = ₹x
Then profits from each part:

​$$\frac{1}{3}x \times \frac{15}{100}$$​​

​$$\frac{1}{4}x \times \frac{20}{100}$$​​

Remaining part = $$1 - \frac{1}{3} - \frac{1}{4} = \frac{5}{12}$$​​

$$→ \frac{5}{12}x \times \frac{24}{100}$$​​

Sum of all profits = ₹62

​$$\frac{1}{3}x \times \frac{15}{100} + \frac{1}{4}x \times \frac{20}{100} + \frac{5}{12}x \times \frac{24}{100} = 62$$​​

​$$x \left( \frac{15}{300} + \frac{20}{400} + \frac{120}{1200} \right) = 62$$​​

​$$x \left( \frac{1}{20} + \frac{1}{20} + \frac{1}{10} \right) = 62$$​​

​$$x \times \frac{4}{20} = 62$$​

​$$x \times \frac{1}{5} = 62$$​

x = $$62 \times 5 =$$​310 (Cost Price of the commodity)

Alternate Method:

Total cost price = Rs.12

1Part                2 part              3 Part

$$\frac 13\times 12 \quad \quad \quad \frac 14\times 12 \quad \quad \quad 5$$​             

4                      3                      5

15%                20%                  24% (profit)

Overall Profit %=$$\frac{60+60+120}{12}$$ = 20%   

Sum of all profits = ₹62​

Cost Price of the commodity = $$62 \times 5 =$$ 310​